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A210448 Total number of different letters summed over all ternary words of length n. 3
0, 3, 15, 57, 195, 633, 1995, 6177, 18915, 57513, 174075, 525297, 1582035, 4758393, 14299755, 42948417, 128943555, 387027273, 1161475035, 3485211537, 10457207475, 31374768153, 94130595915, 282404370657, 847238277795, 2541765165033, 7625396158395, 22876389801777, 68629572058515 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

These are the numbers d(n,3) studied by J. L. Martin. - N. J. A. Sloane, Sep 13 2014

For n >= 0, the number of ternary sequences of length n+1, that contain at least one pair of same consecutive digits. - Armend Shabani, Apr 10 2019

LINKS

Table of n, a(n) for n=0..28.

Philippe Flajolet and Robert Sedgewick, Combinatorial Parameters and MGFs, lecture slides Analytic Combinatorics, 2012

J. L. Martin, The slopes determined by n points in the plane.

Martin, Jeremy L. The slopes determined by n points in the plane arXiv:math/0302106 [math.AG], 2003-2006; Duke Math. J. 131 (2006), no. 1, 119-165. See table of d(n,k), but beware errors.

Index entries for linear recurrences with constant coefficients, signature (5,-6).

FORMULA

E.g.f.: 3*exp(3x) - 3*exp(2x).

  See Mathematica code for a more transparent version of the e.g.f.

  Generally for an m-ary word of length n: m*exp(m*x) - m*exp((m-1)*x)

From Alois P. Heinz, Jan 20 2013: (Start)

a(n) = 3*(3^n-2^n) = 3*A001047(n).

G.f.: 3*x/((3*x-1)*(2*x-1)).

(End)

a(n) = A217764(n,5). - Ross La Haye, Mar 27 2013

EXAMPLE

a(2) = 15 because the length 2 words on alphabet {0,1,2} are: 00, 01, 02, 10, 11, 12, 20, 21, 22 and we sum respectively 1 + 2 + 2 + 2 + 1 + 2 + 2 + 2 + 1 = 15.

MAPLE

a:= n-> 3*(3^n-2^n):

seq(a(n), n=0..30);  # Alois P. Heinz, Jan 20 2013

MATHEMATICA

nn=28; Range[0, nn]!CoefficientList[Series[D[(1+y(Exp[x]-1))^3, y]/.y->1, {x, 0, nn}], x]

(* Second program: *)

LinearRecurrence[{5, -6}, {0, 3}, 30] (* Jean-Fran├žois Alcover, Jan 09 2019 *)

CROSSREFS

Cf. A000918, A001047, A217764.

A diagonal of the triangle in A079268.

Sequence in context: A049187 A049161 A118048 * A284699 A218657 A218804

Adjacent sequences:  A210445 A210446 A210447 * A210449 A210450 A210451

KEYWORD

nonn,easy

AUTHOR

Geoffrey Critzer, Jan 20 2013

STATUS

approved

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Last modified September 16 18:24 EDT 2019. Contains 327116 sequences. (Running on oeis4.)